BALKAN 1987

Let $a$ be a real number and let $f : \mathbb{R}\rightarrow \mathbb{R}$ be a function satisfying $f(0)=\frac{1}{2}$ and $$f(x+y)=f(x)f(a-y)+f(y)f(a-x), \forall x,y \in \mathbb{R}.$$ Prove that $f$ is constant.

Find all real numbers $x,y$ greater than $1$ , satisfying the condition that the numbers $\sqrt{x-1}+\sqrt{y-1}$ and $\sqrt{x+1}+\sqrt{y+1}$ are nonconsecutive integers.

In the triangle $ABC$ the following equality holds: $$\sin^{23}{\frac{A}{2}}\cos^{48}{\frac{B}{2}}=\sin^{23}{\frac{B}{2}}\cos^{48}{\frac{A}{2}}$$ Determine the value of $\frac{AC}{BC}$ .

Two circles $K_{1}$ and $K_{2}$ , centered at $O_{1}$ and $O_{2}$ with radii $1$ and $\sqrt{2}$ respectively, intersect at $A$ and $B$ . Let $C$ be a point on $K_{2}$ such that the midpoint of $AC$ lies on $K_{1}$ . Find the length of the segment $AC$ if $O_{1}O_{2}=2$